Volatility Function Research Articles

Statistical inference for partially linear stochastic models with heteroscedastic errors

Partially linear models are extended linear models where one covariate is nonparametric, which is a good balance between flexibility and parsimony. The partially linear stochastic model with heteroscedastic errors is considered, where the nonparametric part can act as a trend. The estimators of the parametric component, the nonparametric component and the volatility function are proposed. Furthermore, simultaneous confidence bands about the nonparametric part and the volatility function are constructed based on their coverage probabilities, which are shown to be asymptotically correct. By the confidence bands, the problems of hypothesis testing in this model can be solved effectively from a global view. The finite sample performance of the proposed method is assessed by Monte Carlo simulation studies, and demonstrated by the analyses of non-stationary Australian annual temperature anomaly series and non-homoscedastic daily air quality measurements in New York, where the simultaneous confidence bands provide more comprehensive information about the nonparametric and volatility functions.

VIX option pricing and CBOE VIX Term Structure: A new methodology for volatility derivatives valuation

This study integrates CBOE VIX Term Structure and VIX futures to simplify VIX option pricing in multifactor models. Exponential and hump volatility functions with one- to three-factor models of the VIX evolution are used to examine their pricing for VIX options across strikes and maturities. The results show that using exponential volatility functions presents an effective choice as pricing models for VIX calls, whereas hump volatility functions provide efficient out-of-sample valuation for most VIX puts, in particular with deep in-the-money and deep out-of-the-money. Pricing errors for calls can be further reduced with a two-factor model.

Robust Estimation of Semiparametric Multiplicative Volatility Models

We investigate a model in which we connect slowly time varying unconditional long-run volatility with short-run conditional volatility whose representation is given as a semi-strong GARCH (1,1) process with heavy tailed errors. We focus on robust estimation of both long-run and short-run volatilities. Our estimation is semiparametric since the long-run volatility is totally unspecified whereas the short-run conditional volatility is a parametric semi-strong GARCH (1,1) process. We propose different robust estimation methods for nonstationary and strictly stationary GARCH parameters with nonparametric long run volatility function. Our estimation is based on a two-step LAD procedure. We establish the relevant asymptotic theory of the proposed estimators. Numerical results lend support to our theoretical results.

Stochastic volatility for interest rate derivatives

This paper uses an extensive set of market data of forward swap rates and swaptions covering 3 July 2002 to 21 May 2009 to identify a two-dimensional stochastic volatility process for the level of rates. The process is identified step by step by increasing the requirement of the model and introducing appropriate adjustments. The first part of the paper investigates the smile dynamics of forward swap rates at their setting dates. Comparing the SABR (with different ) and Heston stochastic volatility models informs us what different specifications of the driving SDEs have to offer in terms of reflecting the dynamics of the smile across dates. The outcome of the analysis is that a normal SABR model () satisfactorily passes all tests and seems to provide a good match to the market. In contrast, we find that the Heston model does not. The next step is to seek a model of the forward swap rates (in their own swaption measure) based on only two Brownian motions that enables a specification with common parameters. It turns out that this can be done by extending the SABR model with a time-dependent volatility function and a mean-reverting volatility process. The performance of the extended (SABR with mean-reversion) model is analysed over several historical dates and is shown to be a stable and flexible choice that allows for good calibration across expiries and strikes. Finally, a time-homogeneous candidate stochastic volatility process that can be used as a driver for all swap rates is identified. This candidate process may in future work be used as a building block for a separable stochastic volatility LIBOR market model or a stochastic volatility Markov-functional model.

A note on non‐parametric testing for Gaussian innovations in AR–ARCH models

In this paper, we consider autoregressive models with conditional autoregressive variance, including the case of homoscedastic AR models and the case of ARCH models. Our aim is to test the hypothesis of normality for the innovations in a completely non‐parametric way, that is, without imposing parametric assumptions on the conditional mean and volatility functions. To this end, the Cramér–von Mises test based on the empirical distribution function of non‐parametrically estimated residuals is shown to be asymptotically distribution‐free. We demonstrate its good performance for finite sample sizes in a small simulation study.AMS 2010 Classification: Primary 62 M10, Secondary 62 G10

Stable local volatility function calibration using spline kernel

We propose an optimization formulation using the l 1 norm to ensure accuracy and stability in calibrating a local volatility function for option pricing. Using a regularization parameter, the proposed objective function balances calibration accuracy with model complexity. Motivated by the support vector machine learning, the unknown local volatility function is represented by a spline kernel function and the model complexity is controlled by minimizing the 1-norm of the kernel coefficient vector. In the context of support vector regression for function estimation based on a finite set of observations, this corresponds to minimizing the number of support vectors for predictability. We illustrate the ability of the proposed approach to reconstruct the local volatility function in a synthetic market. In addition, based on S&P 500 market index option data, we demonstrate that the calibrated local volatility surface is simple and resembles the observed implied volatility surface in shape. Stability is illustrated by calibrating local volatility functions using market option data from different dates.

Assessing the performance of symmetric and asymmetric implied volatility functions

This study examines several alternative symmetric and asymmetric model specifications of regression-based deterministic volatility models to identify the one that best characterizes the implied volatility functions of S&P 500 Index options in the period 1996–2009. We find that estimating the models with nonlinear least squares, instead of ordinary least squares, always results in lower pricing errors in both in- and out-of-sample comparisons. In-sample, asymmetric models of the moneyness ratio estimated separately on calls and puts provide the overall best performance. However, separating calls from puts violates the put-call-parity and leads to severe model mis-specification problems. Out-of-sample, symmetric models that use the logarithmic transformation of the strike price are the overall best ones. The lowest out-of-sample pricing errors are observed when implied volatility models are estimated consistently to the put-call-parity using the joint data set of out-of-the-money options. The out-of-sample pricing performance of the overall best model is shown to be resilient to extreme market conditions and compares quite favorably with continuous-time option pricing models that admit stochastic volatility and random jump risk factors.

Pricing Basket Options Under Local Stochastic Volatility with Jumps

This paper develops a new approximation formula for pricing basket options in a local-stochastic volatility model with jumps. In particular, the model admits local volatility functions and jump components in not only the underlying asset price processes, but also the volatility processes. To the best of our knowledge, the proposed formula is the first one which achieves an analytical approximation for the basket option prices under this type of the models. Moreover, some numerical experiments confirm the validity of the method.

Let's Get Lade: Robust Estimation of Semiparametric Multiplicative Volatility Models

We investigate a model in which we connect slowly time varying unconditional long-run volatility with short-run conditional volatility whose representation is given as a semi-strong GARCH (1,1) process with heavy tailed errors. We focus on robust estimation of both long-run and short-run volatilities. Our estimation is semiparamentric since the long-run volatility is totally unspecified whereas the short-run conditional volatility is a parametric semi-strong GARCH (1,1) process. We propose different robust estimation methods for nonstationary and strictly stationary GARCH parameters with non-parametric long-run volatility function. Our estimation is based on a two-step LAD procedure. We establish the relevant asymptotic theory of the proposed estimators. Numerical results lend support to our theoretical results.

Unstable volatility: the break-preserving local linear estimator

The objective of this paper is to introduce the break-preserving local linear (BPLL) estimator for the estimation of unstable volatility functions for independent and asymptotically independent processes. Breaks in the structure of the conditional mean and/or the volatility functions are common in Finance. Nonparametric estimators are well suited for these events due to the flexibility of their functional form and their good asymptotic properties. However, the local polynomial kernel estimators are not consistent at points where the volatility function has a break. The estimator presented in this paper generalises the classical local linear (LL). The BPLL estimator maintains the desirable properties of the LL estimator with regard to the bias and the boundary estimation while it estimates the breaks consistently. An extensive Monte Carlo study is shown as well as detailed proofs of the estimator asymptotic behaviour.

Local Volatility Calibration during Turbulent Periods

We propose a methodology to calibrate the local volatility function under a continuous time setting. For this purpose, we used the Markov chain approximation method built on the well-established idea of local consistency. The chain was designed to approximate jump-diffusions coupled with a local volatility function. We found that this method outperforms traditional numerical algorithms that require time discretization. Furthermore, we showed that a local volatility jump-diffusion model outperformed the in- and out-of-sample pricing that the market practitioners benchmark, namely the Practitioners Black-Scholes, in turbulent periods during which at-the-money implied volatilities have risen substantially. As in previous literature concerning local volatility estimation, we represent the local volatility function using a space-time cubic spline.

Pricing Interest Rate Derivatives in a Multifactor HJM Model with Time

We investigate the partial differential equation (PDE) for pricing interest derivatives in the multi-factor Cheyette Model, which involves time-dependent volatility functions with a special structure. The high dimensional parabolic PDE that results is solved numerically via a modified sparse grid approach, that turns out to be accurate and efficient. In addition we study the corresponding Monte Carlo simulation, which is fast since the distribution of the state variables can be calculated explicitly. The results obtained from both methodologies are compared to the known analytical solutions for bonds and caplets. When there is no analytical solution, both European and Bermudan swaptions have been evaluated using the sparse grid PDE approach that is shown to outperform the Monte Carlo simulation.

A Quadratic Volatility Cheyette Model

In this paper we present an extension of the one factor blended Cheyette model for pricing single currency exotics, allowing for a more adequate fit to the swaption volatility smile. We first present a general framework based on the HJM model and then make a separability assumption on the instantaneous forward rate volatility, thus enabling a representation of the discount curve in a finite number of Markovian state variables. We show a practical application of this family of models by analyzing calibration and pricing in the case of a quadratic volatility function. By doing so, we provide a novel and parsimonious specification of the Cheyette model. Then for calibration purposes, we develop fast and accurate approximations for European swaptions, based on standard projection and averaging techniques. We also improve the usual naive mean state estimation by the use of Gaussian approximations. Last we present an efficient large step Monte-Carlo simulation for strongly path dependent exotics.

Non-Parametric Pricing of Interest Rates Options

The pricing models for interest rates derivatives largely used today employ, many times,excessively restrictive premises in regards to the underlying assets' volatility. The Black and Scholes and the Vasicek methods, for example, consider the variance of the series as constant in time and among different maturities, assumption that may not be themost adequate in all cases. In this paper we discuss the non-parametric estimation of the volatility function using a kernel regression technique and later the pricing of options in a Gaussian HJM model. We analyzed different possible specifications for the non-parametric estimation using the Monte Carlo simulations to price options on zero coupon bonds. We also carried out an empirical study using the proposed methodologyfor the pricing of IDI Index options in the Brazilian market..

Estimation of quarticity with high-frequency data

We propose a new methodology based on Fourier analysis to estimate the fourth power of the volatility function (spot quarticity) and, as a byproduct, the integrated function. We prove the consistency of the proposed estimator of the integrated quarticity. Further, we analyse its efficiency in the presence of microstructure noise, from both a theoretical and empirical viewpoint. Extensions to higher powers of volatility and to the multivariate case are also discussed.

A Term Structure Model for VIX Futures

This study develops a term structure model for VIX futures. Instead of deriving the VIX futures price from a model for the instantaneous variance of the S&P 500 or a model for the VIX, the VIX futures price dynamics are specified exogenously. The empirical features of VIX futures returns (positive skewness, excess kurtosis, and a decreasing volatility term structure for longer term expirations) are captured by assuming that they are normal inverse Gaussian distributed and scaled by a volatility function that is dependent on the maturity. The usefulness of the resulting model is illustrated in two applications: risk management (via calculating value at risk (VaR)) and asset pricing (via pricing hypothetical VIX options). The results show that the model provides a good fit for the empirical term structure of VIX futures, produces good VaR estimates, and is promising for use in pricing VIX options. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:421‐442, 2013

Delay geometric Brownian motion in financial option valuation

Motivated by influential work on complete stochastic volatility models, such as Hobson and Rogers [11], we introduce a model driven by a delay geometric Brownian motion (DGBM) which is described by the stochastic delay differential equation . We show that the equation has a unique positive solution under a very general condition, namely that the volatility function V is a continuous mapping from to itself. Moreover, we show that the delay effect is not too sensitive to time lag changes. The desirable robustness of the delay effect is demonstrated on several important financial derivatives as well as on the value process of the underlying asset. Finally, we introduce an Euler–Maruyama numerical scheme for our proposed model and show that this numerical method approximates option prices very well. All these features show that the proposed DGBM serves as a rich alternative in modelling financial instruments in a complete market framework.

A Dynamic Copula Approach to Recovering the Index Implied Volatility Skew

Equity index implied volatility functions are known to be excessively skewed in comparison with implied volatility at the single stock level. We study this stylized fact for the case of a major German stock index, the DAX, by recovering index implied volatility from simulating the 30-dimensional return system of all DAX constituents. Option prices are computed after risk neutralization of the multivariate process which is estimated under the physical probability measure. The multivariate models belong to the class of copula asymmetric dynamic conditional correlation models. We show that moderate tail dependence coupled with asymmetric correlation response to negative news is essential to explain the index implied volatility skew. Standard dynamic correlation models with zero tail dependence fail to generate a sufficiently steep implied volatility skew.

THE HEAT-KERNEL MOST-LIKELY-PATH APPROXIMATION

In this article, we derive a new most-likely-path (MLP) approximation for implied volatility in terms of local volatility, based on time-integration of the lowest order term in the heat-kernel expansion. This new approximation formula turns out to be a natural extension of the well-known formula of Berestycki, Busca and Florent. Various other MLP approximations have been suggested in the literature involving different choices of most-likely-path; our work fixes a natural definition of the most-likely-path. We confirm the improved performance of our new approximation relative to existing approximations in an explicit computation using a realistic S&P500 local volatility function.

Robustifying multivariate trend tests to nonstationary volatility

This article studies inference of multivariate trend model when the volatility process is nonstationary. Within a quite general framework we analyze four classes of tests based on least squares estimation, one of which is robust to both weak serial correlation and nonstationary volatility. The existing multivariate trend tests, which either use non-robust standard errors or rely on non-standard distribution theory, are generally non-pivotal involving the unknown time-varying volatility function in the limit. Two-step residual-based i.i.d. bootstrap and wild bootstrap procedures are proposed for the robust tests and are shown to be asymptotically valid. Simulations demonstrate the effects of nonstationary volatility on the trend tests and the good behavior of the robust tests in finite samples.